# Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras

@article{Oh2019TwistedAF, title={Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras}, author={Se-jin Oh and Uhi Rinn Suh}, journal={Journal of Algebra}, year={2019} }

## 12 Citations

### Continuous Quivers of Type A (II) The Auslander-Reiten Space

- Mathematics
- 2019

This work is the sequel to Continuous Quivers of Type A (I). In this paper we define the Auslander-Reiten space of a continuous type $A$ quiver, which generalizes the Auslander-Reiten quiver of typeβ¦

### Q-data and Representation Theory of Untwisted Quantum Affine Algebras

- MathematicsCommunications in Mathematical Physics
- 2021

For a complex finite-dimensional simple Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}β¦

### PBW theory for quantum affine algebras

- Mathematics
- 2020

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $\mathcal{C}_{\mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl dualityβ¦

### Isomorphisms among quantum Grothendieck rings and propagation of positivity

- Mathematics
- 2022

Abstract Let (π€,π){\mathfrak{g},\mathsf{g})} be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with π{\mathsf{g}} being of simply-lacedβ¦

### Categorical crystals for quantum affine algebras

- MathematicsJournal fΓΌr die reine und angewandte Mathematik (Crelles Journal)
- 2022

Abstract In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B ^ π€ β’ ( β )β¦

### Block decomposition for quantum affine algebras by the associated simply-laced root system

- Mathematics
- 2020

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $\mathscr{C}_\mathfrak{g}$ be the category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules. Weβ¦

### Toroidal Grothendieck rings and cluster algebras

- MathematicsMathematische Zeitschrift
- 2021

We study deformations of cluster algebras with several quantum parameters, called toroidal cluster algebras, which naturally appear in the study of Grothendieck rings of representations of quantumβ¦

### PBW theoretic approach to the module category of quantum affine algebras

- MathematicsProceedings of the Japan Academy, Series A, Mathematical Sciences
- 2021

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $\mathcal{C}^0_{\mathfrak{g}}$ be Hernandez-Leclerc's category. For a duality datum $\mathcal{D}$ inβ¦

### Simply laced root systems arising from quantum affine algebras

- MathematicsCompositio Mathematica
- 2022

Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$, and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrakβ¦

### The (q,Β t)-Cartan matrix specialized at $$q=1$$ q = 1 and its applications

- MathematicsMathematische Zeitschrift
- 2023

The ( q ,Β t )-Cartan matrix specialized at $$t=1$$ t = 1 , usually called the quantum Cartan matrix , has deep connections with (i) the representation theory of its untwisted quantum affine algebra,β¦

## References

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We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using AuslanderβReiten quivers for finite type ADE. Then we can prove that the statisticsβ¦

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We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closelyβ¦

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- 2019

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories $${C_Q^{(t)} (t=1,2,3),β¦

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Abstract We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras: we prove that the characters of Kirillov-Reshetikhin modules solve the Q-system and we give anβ¦

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This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditionsβ¦

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We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtainβ¦

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- Mathematics
- 2011

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of Frenkel and Hernandezβ¦